"Thou" and "thee" were the descendents of Old English "þū" and "þē", respectively, and were originally the nominative and dative[1] forms of the singular second-person pronoun. "Ye" and "you" (< OE "ġē" and "ēow") were the corresponding forms of the plural second-person pronoun. In the High Middle Ages, English adopted the "tu-vous distinction", named after the French pronouns corresponding to "thou-you", wherein social superiors were always addressed in the plural (cf. "the royal we"). "Thou" fell out of use because people did not want to risk offending anyone by implying the other was their social inferior, wherefore they began addressing everyone as a superior.

When Shakespeare writes "I thou thee!", it's not to ironically mock, but rather to explicitly disparage, by asserting that the recipient is inferior, unworthy of being ye'd.

Of course, the vacuum created by the loss of a productive singular was later filled by constructions like "y'all", "you guys", "yous", &c. The same phenomenon occurred in Dutch, where the singular "du"/"di" was replaced by the plural (and therefore formal) "gi/u" during the Middle Dutch period, and the plural later filled by "jullie", a contraction of "je lui", meaning "you folks".

[1]: In late Old English, also the accusative, replacing "þec". From Middle English, the oblique (the sole objective case).

These transitions can also be rapid: my mother-in-law calls me otaku without a hint of irony, since that was a perfectly polite term until it was suddenly adopted to refer to sociopathic geeks in the wake of the Tsutomi Miyazaki murders in 1989. Like many people in her age group, she never got the memo on this.

That's fascinating. In my native language you would have been considered more polite and thou would be for friends or equals.

At the time James VI/I commissioned his Authorized translation, "thou" was already starting to fall out of production, but was not something that seemed foreign or archaic to English speakers, and he'd asked his translators to follow the style of the existing English translations, which were by-and-large based on Tyndale's. There are certainly some ways in which the translation attempts to emphasize certain theological/christological viewpoints, but the usage of "thou" is not one of them.

It wasn't theologically neutral, though. To address God with the polite plural form of the second-person pronoun suggests that the speaker is addressing an entity that occupies a higher position in some kind of social hierarchy, which implies that one's relationship to God is mediated by social structures. Use of the singular form was considered more consistent with Protestant theology, which emphasized a direct and individual connection between the believer and their God.

You see the same phenomenon in German, a language that's retained multiple forms of the second person. In German translations of the Bible, and in German prayers, God is addressed as "Du" (the familiar singular "you").

You also see it in the Latin Vulgate, where God is addressed as "tu" (singular), rather than "vos" (plural). My point is that the translation as "thou" was more-or-less a direct translation of the original text in almost all cases, preserving the distictions made in the Hebrew and Greek manuscripts. At the time those English translations were made, the choice was theologically neutral insofar as the original texts were theologically neutral.